{"id":259498,"date":"2025-07-19T02:41:22","date_gmt":"2025-07-19T02:41:22","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=259498"},"modified":"2025-07-19T02:41:24","modified_gmt":"2025-07-19T02:41:24","slug":"find-the-domain-and-range-of-the-following-piecewise-function","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/19\/find-the-domain-and-range-of-the-following-piecewise-function\/","title":{"rendered":"Find the domain and range of the following piecewise function."},"content":{"rendered":"\n<p>Find the domain and range of the following piecewise function. f(x)=x+4 if -4 &lt; x &lt; 3 2x-1 if 3 &lt; x &lt; 6<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-0-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>To find the domain and range of the piecewise function f(x)={x+4if&nbsp;\u22124&lt;x&lt;32x\u22121if&nbsp;3&lt;x&lt;6f(x) = \\begin{cases} x + 4 &amp; \\text{if } -4 &lt; x &lt; 3 \\\\ 2x &#8211; 1 &amp; \\text{if } 3 &lt; x &lt; 6 \\end{cases}f(x)={x+42x\u22121\u200bif&nbsp;\u22124&lt;x&lt;3if&nbsp;3&lt;x&lt;6\u200b<\/p>\n\n\n\n<p><strong>Domain:<\/strong><\/p>\n\n\n\n<p>The domain of a function is the set of all possible input values (x-values) for which the function is defined. For this function, we have two pieces:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The first piece is x+4x + 4x+4, defined for \u22124&lt;x&lt;3-4 &lt; x &lt; 3\u22124&lt;x&lt;3.<\/li>\n\n\n\n<li>The second piece is 2x\u221212x &#8211; 12x\u22121, defined for 3&lt;x&lt;63 &lt; x &lt; 63&lt;x&lt;6.<\/li>\n<\/ul>\n\n\n\n<p>The function is not defined at x=3x = 3x=3, since the interval \u22124&lt;x&lt;3-4 &lt; x &lt; 3\u22124&lt;x&lt;3 doesn&#8217;t include 333, and the second interval 3&lt;x&lt;63 &lt; x &lt; 63&lt;x&lt;6 starts just after 333. So, the domain is the union of these two intervals:<\/p>\n\n\n\n<p><strong>Domain:<\/strong> (\u22124,3)\u222a(3,6)(-4, 3) \\cup (3, 6)(\u22124,3)\u222a(3,6)<\/p>\n\n\n\n<p><strong>Range:<\/strong><\/p>\n\n\n\n<p>The range of a function is the set of all possible output values (y-values). We will determine the range of each piece separately and then combine them.<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>For the first piece, f(x)=x+4f(x) = x + 4f(x)=x+4, where \u22124&lt;x&lt;3-4 &lt; x &lt; 3\u22124&lt;x&lt;3:<ul><li>The smallest value of xxx is just greater than \u22124-4\u22124, so f(x)f(x)f(x) is just slightly greater than \u22124+4=0-4 + 4 = 0\u22124+4=0.<\/li><li>The largest value of xxx is just less than 333, so f(x)f(x)f(x) is just slightly less than 3+4=73 + 4 = 73+4=7.<\/li><\/ul>Hence, the range of the first piece is (0,7)(0, 7)(0,7).<\/li>\n\n\n\n<li>For the second piece, f(x)=2x\u22121f(x) = 2x &#8211; 1f(x)=2x\u22121, where 3&lt;x&lt;63 &lt; x &lt; 63&lt;x&lt;6:<ul><li>The smallest value of xxx is just greater than 333, so f(x)f(x)f(x) is just slightly greater than 2(3)\u22121=52(3) &#8211; 1 = 52(3)\u22121=5.<\/li><li>The largest value of xxx is just less than 666, so f(x)f(x)f(x) is just slightly less than 2(6)\u22121=112(6) &#8211; 1 = 112(6)\u22121=11.<\/li><\/ul>Hence, the range of the second piece is (5,11)(5, 11)(5,11).<\/li>\n<\/ol>\n\n\n\n<p><strong>Range:<\/strong><\/p>\n\n\n\n<p>The overall range is the union of the two ranges: (0,7)\u222a(5,11)(0, 7) \\cup (5, 11)(0,7)\u222a(5,11)<\/p>\n\n\n\n<p><strong>Final Answer:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Domain:<\/strong> (\u22124,3)\u222a(3,6)(-4, 3) \\cup (3, 6)(\u22124,3)\u222a(3,6)<\/li>\n\n\n\n<li><strong>Range:<\/strong> (0,7)\u222a(5,11)(0, 7) \\cup (5, 11)(0,7)\u222a(5,11)<\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/07\/learnexams-banner6-980.jpeg\" alt=\"\" class=\"wp-image-259499\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Find the domain and range of the following piecewise function. f(x)=x+4 if -4 &lt; x &lt; 3 2x-1 if 3 &lt; x &lt; 6 The Correct Answer and Explanation is: To find the domain and range of the piecewise function f(x)={x+4if&nbsp;\u22124&lt;x&lt;32x\u22121if&nbsp;3&lt;x&lt;6f(x) = \\begin{cases} x + 4 &amp; \\text{if } -4 &lt; x &lt; 3 \\\\ [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-259498","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/259498","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=259498"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/259498\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=259498"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=259498"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=259498"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}